Question

The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

• A. $(-1{)}^k(k-1)\pi$
• B. $(-1{)}^{k-1}(k-1)\pi$
• C. $(-1{)}^{k}k\pi$
• D. $(-1{)}^{k-1}k\pi$

Answer 1

The correct option is A $(-1{)}^k(k-1)\pi$

$f^{\prime }(k-0)={lim}_{h\to 0}\frac{[k-h]sin\pi (k-h)-\left[k\right]sin\pi k}{-h}$
$={lim}_{h\to 0}\frac{(-1{)}^{k-1}(k-1)sin\pi h-k\times 0}{-h}$
$={lim}_{h\to 0}\frac{(-1{)}^{k-1}(k-1)sin\pi h}{-h}=(-1{)}^k.(k-1)\pi$